In this paper we construct ultraproducts of admissible models and use them to derive compactness theorems that combine with completeness to yield strong completeness: any QS-consistent set of formulas is satisfiable in a model whose admissible propositions validate S.
The Barcan Formula is analysed separately and shown to axiomatise certain logics that are strongly complete over admissible models in which the quantifiers are given their standard Kripkean interpretation.